Best Known (196, 196+45, s)-Nets in Base 3
(196, 196+45, 896)-Net over F3 — Constructive and digital
Digital (196, 241, 896)-net over F3, using
- 3 times m-reduction [i] based on digital (196, 244, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 61, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 61, 224)-net over F81, using
(196, 196+45, 3845)-Net over F3 — Digital
Digital (196, 241, 3845)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3241, 3845, F3, 45) (dual of [3845, 3604, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(3241, 6562, F3, 45) (dual of [6562, 6321, 46]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,22], and minimum distance d ≥ |{−22,−21,…,22}|+1 = 46 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3241, 6562, F3, 45) (dual of [6562, 6321, 46]-code), using
(196, 196+45, 725720)-Net in Base 3 — Upper bound on s
There is no (196, 241, 725721)-net in base 3, because
- 1 times m-reduction [i] would yield (196, 240, 725721)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 3 229308 605851 124038 673707 788440 434849 041146 189065 528701 760198 793983 414406 118852 571761 148945 324133 645259 384130 565769 > 3240 [i]